An example of traditional methods for improving image quality is described in Non-Patent Document 1. The image quality improving method restores a higher resolution image on a subject from a plurality of input images in which the same subject is taken so that positions of the subject are shifted by sub-pixel unit.
Assume that there are N input images (gn) (0≦n≦N−1). In the method described in Non-Patent Document 1, each input image (gn) is regarded to be image taken by going through the image taking process expressed by the following equation.gn(x,y)=s↓(h(u,v)*f−(Tn(x,y)))+ηn(x,y)  (1)
Here, f− is a high resolution image of a subject, Tn is a geometric conversion for nth input image, h is a point spread function which is invariant and linear with respect to any coordinate (x, y) on the image, s↓ is an operator for down-sampling the image, and ηn is noise expressed by a normal distribution with mean zero. The above equation can be expressed in matrix operation form as the equation (2).
[Equation 1]gn=Mn f+ηn  (2)
The f− in the equation (2) is a lexicographic ordering of a pixel value f−(x, y). The same goes for the gn and ηn in the equation (2). Also, Mn in the equation (2) is a single matrix into which Tn, h, and s↓ in the equation (1), each of which is linear transformation, are combined.
The image taking process of all the input images can be put into a single equation to obtain the equation (3).
                    [                  Equation          ⁢                                          ⁢          2                ]                                                                                  [                                                                                g                    0                                                                                                                    g                    1                                                                                                ⋮                                                                                                  g                                          N                      -                      1                                                                                            ]                    =                                                    [                                                                                                    M                        0                                                                                                                                                M                        1                                                                                                                        ⋮                                                                                                                          M                                                  N                          -                          1                                                                                                                    ]                            ⁢                              f                _                                      +                          [                                                                                          η                      0                                                                                                                                  η                      1                                                                                                            ⋮                                                                                                              η                                              N                        -                        1                                                                                                        ]                                      ⁢                                  ⁢                              or            ⁢                                                  ⁢            g                    =                                    M              ⁢                              f                _                                      +            η                                              (        3        )            
At this equation, the maximum a posterior estimate (fmap) of the high resolution image can be expressed as the equation (4).
[Equation 3]fmap=argfmax−λ∥f−favg∥2−∥Mf−g∥2  (4)
In this regard, favg is an average image for which the positions in each input image are adjusted to be aligned.
In order to solve this problem, a numerical calculation technique such as a conjugate gradient method or the like is generally used. That is, starting the calculation from a certain initial value, an optimum solution can be obtained by converging solutions with performing an iterative calculation.
Non-Patent Document 1: D. Capel “Image Mosaicing and Super-Resolution”, Springer Verlag, January, 2004, pp. 86-147